A function with no max or min at an endpoint

In summary, the conversation discusses the concept of extreme values in functions. It is mentioned that although a function cannot have extreme values at points other than endpoints, critical points, and singular points, it is possible for a function to have no extreme values at these points. An example from "Calculus: A Complete Course" is given as reference. The conversation also touches upon the difficulty of graphing a function with an endpoint where it fails to have an extreme value, with an example of a sine function given. The conversation then turns to discussing different methods to avoid local extreme values at endpoints, such as using oscillating functions or the Weierstrass function. The conversation concludes with the person expressing their confusion and need for further clarification on the
  • #1
mcastillo356
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TL;DR Summary
I've found an exercise, and need to revisit the concepts involved, first of all, and eventually solve it. I have some clues, but at the same time, a need to start from zero.
Hi, PF

Although a function cannot have extreme values anywhere other than at endpoints, critical points, and singular points, it need not have extreme values at such points. There is an example of how a function need not have extreme values at a critical point or a singular point in 9th edition of "Calculus: A Complete Course", pg 237, Figure 4.18. It is more difficult to draw the graph of a function whose domain has an endpoint at which the function fails to have an extreme value. See Exercise 49 at the end of this section for an example of such a function.
I would like to know why ##x=0## is an endpoint of such function, and why does it fail to have an extreme value, for the first instance. My aim is to completely understand and solve it.

Greetings, Love!
 
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  • #2
Sorry, lack of statement of the exercise, and clues to solve it I have
Now have no time. Let's leave for tomorrow
Love
 
  • #3
It's not hard to draw such a function. Graph the sine function. Pick an interval that encloses the maximum and the minimum but goes beyond them to points that are neither a max or min. ##[0,\pi]## will do.
 
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  • #4
Fine. Think the example I will put is going to be much more complicated. At first glance.
See later
 
  • #5
If you are talking about local maximums and minimums, it is harder to avoid those at the endpoints. But it can be done.
 
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  • #6
Copying the function would have been useful for users who don't have that book.

You can avoid a local minimum/maximum at an endpoint using oscillating functions for example. ##f(x)=x \sin(\frac 1 x)## for ##x>0##, ##f(x)=0## is continuous but oscillates in the ##\pm x## "cone" infinitely often as you approach x=0. If the function doesn't need to be continuous there is even more freedom how to do that.
You simply define the interval to end at x=0. It's arbitrary which range to look at.
 
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  • #8
Mark44 said:
There's always the Weierstrass Function (https://en.wikipedia.org/wiki/Weierstrass_function) that is continuous everywhere, but nowhere differentiable.
I've taken a glance. Very advanced to me
mfb said:
Copying the function would have been useful for users who don't have that book.

You can avoid a local minimum/maximum at an endpoint using oscillating functions for example. ##f(x)=x \sin(\frac 1 x)## for ##x>0##, ##f(x)=0## is continuous but oscillates in the ##\pm x## "cone" infinitely often as you approach x=0. If the function doesn't need to be continuous there is even more freedom how to do that.
You simply define the interval to end at x=0. It's arbitrary which range to look at.
Thanks, this is fine, it is what I was going to post.
This is going to be hard work for me. This island I hope will once again support my lack of rigor and inconsistency. I'm also in touch with "Rincón Matemático", and some of you will have noticed that at my first post I made a question I should already have known the answer to. In my confusing paragraph I say:
mcastillo356 said:
I would like to know why ##x=0## is an endpoint of such function
This is something I should know from

https://www.physicsforums.com/threads/what-kind-of-definition-is-this.1008348/

Where I said I had understood the definition, and thus, the concept. Well, still wondering how to face:

-The definition at my previous post: I'm thinking now about personal classes.
-The exercise proposed here: I'm thinking about PF, the Spanish Forum, a personal teacher.

Well, too many words, just to apologize about my first post, and to introduce at last my wonders about this thread's direction. Still don't made the first step: write the question and contribution or effort to solve it. This is just a statement of intent, an outline of what i wish was the thread, and a plea: I need time to draft properly all I have in mind.

Thanks! Love
 

FAQ: A function with no max or min at an endpoint

What does it mean for a function to have no max or min at an endpoint?

When a function has no max or min at an endpoint, it means that the function does not have a highest or lowest value at the beginning or end of its domain.

How can a function have no max or min at an endpoint?

This can occur when the function has a horizontal asymptote at the endpoint, meaning that the function approaches a certain value but never reaches it.

What is the significance of a function having no max or min at an endpoint?

This means that the function has no local extrema at the endpoint, which can affect the behavior and shape of the function.

Can a function still have a global max or min if it has no max or min at an endpoint?

Yes, it is possible for a function to have a global max or min even if it does not have a max or min at an endpoint. This can occur if the function has a local max or min within its domain.

How does the absence of a max or min at an endpoint affect the continuity of a function?

Having no max or min at an endpoint does not necessarily affect the continuity of a function. A function can still be continuous even if it does not have a max or min at an endpoint.

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